The three-dimensional (3D) Dirac point, where two Weyl points overlap in momentum space, is usually unstable and hard to realize. Here we show, based on the first-principles calculations and effective model analysis, that crystalline A 3 Bi (A=Na, K, Rb) are Dirac semimetals with bulk 3D Dirac points protected by crystal symmetry. They possess non-trivial Fermi arcs on the surfaces, and can be driven into various topologically distinct phases by explicit breaking of symmetries. Giant diamagnetism, linear quantum magnetoresistance, and quantum spin-Hall effect will be expected for such compounds. PACS numbers: 71.20.-b, 73.20.-r, 73.43.-t 1 arXiv:1202.5636v2 [cond-mat.mtrl-sci]

In 3D momentum space, a topological phase boundary separating the Chern insulating layers from normal insulating layers may exist, where the gap must be closed, resulting in a "Chern semimetal" state with topologically unavoidable band crossings at the fermi level. This state is a condensedmatter realization of Weyl fermions in (3+1) D, and should exhibit remarkable features, such as magnetic monopoles and fermi arcs. Here we predict, based on first-principles calculations, that such a novel quantum state can be realized in a known ferromagnetic compound HgCr2Se4, with a single pair of Weyl fermions separated in momentum space. The quantum Hall effect without an external magnetic field can be achieved in its quantum-well structure.PACS numbers: 71.20.-b, 73.20.-r, 73.43.-f Under broken time reversal symmetry, the topological phases of two-dimensional (2D) insulators can be characterized by an integer invariant, called Chern number [1], which is also known as the TKNN number [2] or the number of chiral edge states [3] in the context of the quantum Hall effect. 2D insulators can thus be classified as normal insulators or Chern insulators depending on whether or not the Chern number vanishes. Since the Chern invariant is defined only for 2D insulators, it is natural to ask what is its analog in 3D? Starting from a 2D Chern insulating plane (say at k z = 0), and considering its evolution as a function of k z , generally two situations may happen. If the dispersion along k z is weak, such that the Chern number remains unchanged, the system can be viewed as the simple stacking of 2D Chern insulating layers along the z direction. Such 3D Chern insulators are trivial generalization of the Chern number to 3D, which is quite similar to the weak topological insulators in systems with time reversal symmetry. However, if the dispersion along k z is strong, such that Chern number changes as the function of k z , the system will be in a nontrivial semimetal state with "topologically unavoidable" band crossings located at the phase boundary separating the insulating layers in k space with different Chern numbers [4,5]. This is due to the fact that the change of Chern number corresponds to a topological phase transition, which can happen only if the gap is closed. From the Kohn-Luttinger theorem, we can always expect that the band crossings appear at the fermi level at stoichiometry. This Chern semimetal state, if found to exist, can be regarded as a condensed matter realization of (3+1)D chiral fermions (or called Weyl fermions) in the relativistic quantum field theory, where the field can be described by the 2-component Weyl spinors [6] (either left-or righthanded), which are half of the Dirac spinors and must appear in pairs. The band-crossing points or Weyl nodes are topological objects in the following senses. First, since no mass is allowed in 2×2 Hamiltonian, the Weyl nodes should be locally stable and can only be removed when a pair of Weyl nodes meet together in the k space. Second the Weyl nodes are "topological def...

The search for large-gap quantum spin Hall (QSH) insulators and effective approaches to tune QSH states is important for both fundamental and practical interests. Based on first-principles calculations we find two-dimensional tin films are QSH insulators with sizable bulk gaps of 0.3 eV, sufficiently large for practical applications at room temperature. These QSH states can be effectively tuned by chemical functionalization and by external strain. The mechanism for the QSH effect in this system is band inversion at the Γ point, similar to the case of a HgTe quantum well. With surface doping of magnetic elements, the quantum anomalous Hall effect could also be realized.